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研究生：	張瑞航 Chang, Jui-Hang
論文名稱：	量子布朗運動的精確主方程式 An Exact Master Equation for a Quantum Brownian Particle
指導教授：	楊緒濃 Nyeo, Su-Long
學位類別：	碩士 Master
系所名稱：	理學院 - 物理學系 Department of Physics
論文出版年：	2006
畢業學年度：	94
語文別：	英文
論文頁數：	75
中文關鍵詞：	開放式量子系統、影響泛函、非馬可夫過程
外文關鍵詞：	open quantum system, influence functional, non-Markovian processes
相關次數：	點閱：87 下載：4
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我們利用 Feynman 和 Vernon 發展出的影響泛函方法來研究系統加上熱庫的模型＿即在任意溫度下，一個在諧振位勢中的粒子與一般環境 (歐姆, 次歐姆與超歐姆) 線性耦合的模型。我們推導出系統約化密度矩陣的精確主方程式，並對主方程式中與時間相關的係數 (耗散係數，擴散係數，…) 做了弱耦合近似。我們也對這些與時間相關的係數，在各種不同的環境中進行數值分析。值得注意的是，系統約化密度矩陣的快速對角化發生在極低的環境溫度中，而且發生的時間尺度也比在期刊論文　Phys. Rev. D 45, 2843 (1992) 中所分析的較強耦合條件下的尺度來的長。

We apply the Feynman-Vernon influence functional method to study the system-plus-reservoir model－
a Brownian particle in a harmonic potential　linearly coupled to a general environment (ohmic, sub-ohmic, or supraohmic) at arbitrary temperature. We derive an exact master equation for the reduced density matrix of the system and then make a weak-coupling approximation for the time-dependent coefficients (dissipation constant, diffusive coefficient, ...) in the master equation. We also analyze numerically these time-dependent coefficients for different environments. Noteworthily, the fast diag-
onalization of the reduced density matrix takes place at a very low temperatrue and in a time scale larger than the one in the case of stronger coupling analyzed in Hu's paper, Phys. Rev. D 45, 2843 (1992).

Contents
1 Quantum Brownian Motion: An Introduction                                1
  1.1 Classical Motion of a Brownian Particle . . . . . . . . . . . . . . 1
  1.2 Approaches to Open Quantum Systems . . . . . . . . . . . . . . . .  3
      1.2.1 Quantum Langevin Equation for a Brownian Particle . . . . . . 4
      1.2.2 The Ohmic, Subohmic, and Supraohmic Environment . . . . . . . 9
      1.2.3 Commutation Relations for the Noise Operator and its Statis-
            tics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
      1.2.4 From Quantum Langevin Equation to Master Equation . . . . .  14
2 Review of Feynman's Space-Time Formulation of Nonrelativistic Quantum 
  Mechanics                                                              21
  2.1 The Single Path Integrals . . . . . . . . . . . . . . . . . . . .  21
      2.1.1 The Action and the Action of Classical Path . . . . . . . .  21
      2.1.2 The Quantum-Mechanical Amplitude K(xb;xa) and Single Path
            Integrals . . . . . . . . . . . . . . . . . . . . . . . . .  22
      2.1.3 Gaussian Integrals . . . . . . . . . . . . . . . . . . . . . 23
  2.2 The System with Many Variables and Double Path Integrals . . . . . 24
      2.2.1 The Feynman's Space-Time Formulation with Influence Functional
            Method [7] . . . . . . . . . . . . . . . . . . . . . . . . . 25
      2.2.2 The General Properties of the Influence Functional . . . . . 26
      2.2.3 Quantum Statistical Mechanics . . . . . . . . . . . . . . .  27
      2.2.4 The Operator Form of Influence Functional . . . . . . . . .  28
      2.2.5 Non-Markovian Behavior of the Reduced Evolution Operator . . 29
3 Non-Markovian Dynamics of Quantum Brownian Motion in a General Environ-
  ment                                                                   31
  3.1 The System and Environment . . . . . . . . . . . . . . . . . . . . 31
      3.1.1 The Complex Kernel L(s) and the Interpretation of the Noise 
            and Dissipation Kernels . . . . . . . . . . . . . . . . . .  34
      3.1.2 Decoherence Versus Dissipation: Ohmic Environment at High  
            Temperatures . . . . . . . . . . . . . . . . . . . . . . . . 36
  3.2 The Master Equation in a General Environment . . . . . . . . . . . 37
  3.3 Analysis of the Time-Dependent Coefficients for the Weak-Coupling  
      Approximation . . . . . . . . . . . . . . . . . . . . . . . . . .  47
      3.3.1 The Weak-Coupling Limit . . . . . . . . . . . . . . . . . .  47
      3.3.2 The Frequency Shift . . . . . . . . . . . . . . . . . . . .  47
      3.3.3 The Dissipative Constant . . . . . . . . . . . . . . . . . . 50
      3.3.4 The Diffusive Coefficient and Decoherence . . . . . . . . .  52
4 Discussions and Conclusions                                            56
A Evaluations of the Quantum Mechanical Average for the Noise Operator   57
B Evaluation of the Influence Functional F[x;x']                         59
  B.1 The Density Matrix for the nth Oscillator . . . . . . . . . . . .  59
  B.2 The Kernel K(n)x . . . . . . . . . . . . . . . . . . . . . . . . . 62
  B.3 The Influence Functional F(n)[x;x'] . . . . . . . . . . . . . . .  64
C The Derivation of the Master Equation                                  66
  C.1 The Expansions of Jr(t+dt;0) under the Straight-Path Approximation 66
  C.2 The Equations of Motion for classical paths . . . . . . . . . . .  69
  C.3 The Fresnel Integrals . . . . . . . . . . . . . . . . . . . . . .  71
  C.4 Some Useful Relations . . . . . . . . . . . . . . . . . . . . . .  72
                                    

[1] L. E. Reichl, A Modern Course in Statistical Physics, 2nd ed (John Wiley &
Sons, Inc., New York, 1998).
[2] C. W. Gardiner and P. Zoller, Quantum Noise, 3rd ed (Springer-Verlag,
Berlin, 2004).
[3] U. Weiss, Quantum Dissipative System, 2nd ed (World Scientific, Singapore,
1999)
[4] A. J. Leggett et al., Rev. Mod. Phys. 59, 1 (1987).
[5] A. O. Caldeira and A. J. Leggett, Ann. Phys. (N.Y.) 149, 374 (1983).
[6] I. S. Gradshteyn and I. M. Rhyzhik, Tables of Integrals, Series, and
Products (Academic Press, New York, 1996, 3.987 ETI88(3) and 3.893 BI
((261))(3).)
[7] R. P. Feynman and F. L. Vernon, Ann. Phys. (N.Y.) 24, 118 (1963).
[8] B. L. Hu, Juan Pablo Paz, and Yuhong Zhang, Phys. Rev. D 45, 2843 (1992).
[9] A. O. Caldeira and A. J. Leggett, Physica A 121, 587 (1983).
[10] A. Stern, Y. Aharonov, and Y. Imry, Phys. Rev. A 41, 3436 (1990).

校內：2007-06-22公開
校外：2007-06-22公開

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